words - What is the exact definition of Shituf (vis a vis 'Avodah Zarah)?

"Shituf" is the heretical idea of some sort of co-mingling of something with G-d (Has VeShalom). According to many rabbinic opinions, Christianity is Shituf (cf., Is Christianity Avodah Zara?), and there are differences of opinion as to whether or not this is permitted for non-Jews.

Is Shituf:

1. a belief that something else has the powers of G-d along with G-d;


2. a belief that something else is worthy of worship along with G-d, although G-d is the Almighty;


3. a belief that something else IS (Has VeShalom) G-d, but G-d is G-d as we believe;


4. something else other than I've mentioned above; or


5. some combination of the above?

Answer

Literally, Shituf, שתף, means sharing, and one who shares ownership is called a Shotef, שותף , partner.

The rules defining Shituf-Hashem could be the same ones that define a business-Shotef.

Sanhedrin 63b, Tosfot's comment on the type Shituf that a Jew is allowed to let a non-Jew make, is a comment directly on the prohibition of forming a business-partnership (shotef) with a non-Jew. Here, the term "Shituf" is used interchangeably for the business-parntership (שותף) and the oath a non-Jew makes to a partner of Hashem (משותף):

Sanhedrin 63b,

"One must not form a Shotef [business partnership] [with a non-Jew]"

... "Even though they associate (shituf) God's name with "something else", we do not find that it is forbidden to cause others to associate (shituf)" wikipedia's translation

So, if we want to know what amount of co-mingling is considered "Shituf" or "sharing" when we apply it to Hashem, then I think we can accurately define Shituf by looking how the Talmud identifies what percentage of sharing-ownership (Shotef) of a first-born-calf with a non-Jew is enough of a percentage to render the first-born-calf unnecessary to be redeem.

Bechoros 2b and Sanhedrin 63b use the same quote from "The Father of Samuel". So Bechoros 2b is definitely discussing the same exact business-partnership and oaths as Sanhedrin 63b

bechoros 2b:

"The Father of Samuel" said: One must not form a partnership (שותף) with a non-Jew lest he will be bound to take an oath to him and he will swear in the name of his idol. ...

But the Sages say, so long as a gentile has a share in it (lit. has a hand in the middle - any amount), it is exempt from the law of first-born.

Based on this, I think Shituf would include all of your examples and more.

It is any concept where Hashem shares any percent of his power with another entity/partner. Whether Hashem is 99.99% God and another entity shares .01%, or if Hashem has .01% and the other entity is 99.99% God, or anything between there (has a hand in the middle). (ie. anything that violates "Hashem is one")

Avodah Zarah, on the other hand takes the next step of either making a physical representation to worship and emparting powers on something physical (eg. image, stone, wood, nature, bones), or saying Hashem isn't God at all, something else is (spiritually or physically). (ie. anything that violates "Hashem is God")

(or any of these practices of people who worship Avodah Zarah)

EDIT: Anything above on Sanhedrin 63b.